Answer
please see step by step.
Work Step by Step
A B C D E F G H I....
1 .2 3 .4 5 6 .7 8 9...
$L,O,V,E \leftrightarrow 12,15,22,5$
Encoding with A:
$\left[\begin{array}{ll}
4 & -1\\
-3 & 1
\end{array}\right]\left[\begin{array}{l}
12\\
15
\end{array}\right]=\left[\begin{array}{l}
48-15\\
-36+15
\end{array}\right]=\left[\begin{array}{l}
33\\
-21
\end{array}\right]$
$\left[\begin{array}{ll}
4 & -1\\
-3 & 1
\end{array}\right]\left[\begin{array}{l}
22\\
5
\end{array}\right]=\left[\begin{array}{l}
88-5\\
-66+5
\end{array}\right]=\left[\begin{array}{l}
83\\
-61
\end{array}\right]$
LOVE is encoded into $33,-21,83,-61.$
Decoding with $A^{-1}$
$\left[\begin{array}{ll}
1 & 1\\
3 & 4
\end{array}\right]\left[\begin{array}{l}
33\\
-21
\end{array}\right]=\left[\begin{array}{l}
33-21\\
99-84
\end{array}\right]$
$=\left[\begin{array}{l}
12\\
15
\end{array}\right] \quad...\left(\begin{array}{l}
L\\
O
\end{array}\right)$
$\left[\begin{array}{ll}
1 & 1\\
3 & 4
\end{array}\right]\left[\begin{array}{l}
83\\
-61
\end{array}\right]=\left[\begin{array}{l}
83-61\\
249-244
\end{array}\right]$
$=\left[\begin{array}{l}
22\\
5
\end{array}\right] \quad...\left(\begin{array}{l}
V\\
E
\end{array}\right)$
